Ferromagnetic Luttinger Liquids And: an Exact Integral Equation for the Renormalized Fermi Surface Peter Kopietz

Ferromagnetic Luttinger liquids and: An exact integral equation for the renormalized Fermi surface Peter Kopietz

Institut fur¨ Theoretische Physik, Universit¨at Frankfurt am Main

Lorenz Bartosch, Marcus Kollar, and Peter Kopietz, Phys. Rev. B 67, 092403 (2003). Sascha Ledowski and Peter Kopietz, J.Phys.: Condens. Matter 15, 4779 (2003).

1/22 Introduction

Can itinerant electrons in 1d form a ferromagnetic ground state? • Experiments and numerical studies indicate: • ferromagnetic ground state is possible in 1d! here: use perturbation theory and bosonization • to calculate correlation functions (spin correlations, single-particle Green function) Symmetry-broken Luttinger liquid (analogous to weak ferromagnets in 3d) • what happens to spin-waves in symmetry broken phase? • anomalous dimension close to quantum phase transition can be large! •

2/22 Lieb-Mattis theorem, (1962) There is no ferromagnetic ground state in a 1d conductor!? valid under fairly general conditions: spin and velocity independent forces • N pˆ2 Hˆ = + V (xˆ , . . . , xˆ ) • 2m∗ 1 N Xi=1 lattice model with nearest neighbor hopping • example: N=2: Helium atom (in any dimension) • E(S = 0) < E(S = 1)

3/22 Evidence for 1d itinerant ferromagnetism

“0.7 structure” of quantum • point contacts has been linked to spontaneous ferromagnetism (Thomas et al., 1996)

Extended Hubbard model can show 1d ferromagnetism (Daul and Noack, • PRB 1998) DFT calculation predicts band ferromagnetism for purely organic polymers • (Arita et al., PRL 2002).

4/22 Interacting electrons in 1d

1 Hamiltonian Hˆ = cˆ† cˆ + [f ρˆ ( q)ρˆ (q) + f ρˆ ( q)ρˆ (q)] k kσ kσ 2L n n − n m n − n Xkσ Xq

ρˆ (q) = cˆ† cˆ (charge density) • n kσ kσ k+qσ P ρˆ (q) = σcˆ† cˆ (spin density) • m kσ kσ k+qσ for weak Pferromagnets (m k /π): expand energy dispersion close to • F Fermi points:

(k k )2 λ = + v (k k ) + − F + (k k )3 k kF F − F 2m∗ 6 − F cubic parameter λ will be necessary to stabilize ferromagnetic ground state • Hˆ is spin-rotationally invariant •

5/22 A ﬁrst step: Hartree-Fock theory write Hamiltonian as Hˆ = Hˆ0 + Hˆ1 Hartree-Fock part:

L Hˆ µNˆ = ξ cˆ† cˆ [f n2 + f m2] , ξ = µ + σ∆ , ∆ = f m 0 − kσ kσ kσ − 2 n m kσ k − − m Xkσ

ﬂuctuation part:

1 Hˆ = f δρˆ ( q)δρˆ (q) , δρˆ (q) = ρˆ (q) δ ρˆ (0) 1 2L i i − i i i − q,0h i i Xq,i self-consistent determination of k↑, k↓, m, δn = n(m) n(0) at constant µ: − + f δn = σ∆ kσ − kF n δn = π−1(k + k 2k )  k , k , δn, m ↑ ↓ F  ↑ ↓ −1 −  ⇒ m = π (k↑ k↓) − 6/22  Change in ground-state energy at constant m

Hartree Fock ground state energy Ω(m) = Hˆ0 µNˆ Landau expansion for weak ferromagnetismhπm− k i F

Lv A Ω(m) Ω(0) = F I (I 1)(πm)2 + I4(πm)4 + . . . − 4π − 0 0 − 4 0

0 < I 0 < 1 m)

δΩ( I 0 = 1

I 0 > 1 -m0 m m0

Stoner-parameter: I = 2f /πv 0 − m F quartic coeﬃcient: A = 1 [λ 3λ2/(1 + F )] , λ = (m∗v )−1 , λ = λ/v 12 2 − 1 0 1 F 2 F non-trivial solution at πm = [2(I 1)/(I3A)]1/2 if I > 1. 0 0 − 0 0 7/22 Longitudinal spin ﬂuctuations in RPA

RPA for longitudinal spin- and charge susceptibilities (f = f = f > 0): n − m 0 χRPA(q, iω) = [χ0 + χ0 4f χ0 χ0 ]/D , nn ↑↑ ↓↓ − 0 ↑↑ ↓↓ RPA 0 0 0 0 χmm (q, iω) = [χ↑↑ + χ↓↓ + 4f0χ↑↑χ↓↓]/D ,

where D(q, iω) = 1 4f 2χ0 χ0 and − 0 ↑↑ ↓↓

0 1 f(ξk+q/2,σ0 ) f(ξk−q/2,σ) χσσ0 (q, iω) = − −L ξ 0 ξ iω Xk k+q/2,σ − k−q/2,σ − For small q and ω:

2 0 vσ q χσσ(q, iω) 2 2 ≈ π (vσq) + ω

2 1 λ1 2 vσ/vF = 1 + σλ1∆/vF + λ2 (∆/vF ) + . . . 2 − 1 + F0 8/22 ballistic longitudinal spin ﬂuctuations

Wanted: correlation functions: 1 S (x, t) = [ ρˆ (x, t)ρˆ (0, 0) ρˆ (x, t) ρˆ (0, 0) ] i 2πV h i i i − h i ih i i

−1 Fourier-transform: dynamic structure factor: Si(q, ω) = π Imχii(q, ω + i0)

Within RPA: longitudinal spin ﬂuctuations propagate ballistically!

SRPA(q, ω) = Z q δ(ω v q ) no Landau damping in 1d! • m m| | − m| | weight: Z = 1/π√δ 1 • m 0 velocity v = v √δ v • m F 0 F δ 2(I 1)/I 1 measures distance to zero temperature critical point • 0 ≡ 0 − 0 (quantum phase transition) susceptibility sum rule: 2 lim ∞ dω SRPA(q, ω) = χ • q→0 0 ω m m spin susceptibility: χ−1 = L−1 ∂2RΩ(m)/∂m2 m m0

9/22 transverse suscrptibility: ladder approximation ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¢£¤¥ ¦§ ¡ ¡ ¡ ¡ ¡ LAD LAD 0 −1 −1 ¡ ¡ ¡ ¡ ¡ χ (q, iω) [χ (q, iω) 2f ] ¡ ¡ ¡ ¡ ↑↓¡ ↑↓ 0 ¡ ¡ ¡ ¡ ¡

¡ ¡ ¡ ¡ ≡¡ − ¡ ¡ ¡ ¡ ¡ for small q and ω:

m iω χ0 (q, iω) 0 1 + Bq2 ↑↓ ≈ 2∆ 2∆ −

−1 −1 k↑ 2 1 2 B = [4∆(k↑ k↓)] [v↑ + v↓ ∆ dkv ] = [λ2 λ ] − − k↓ k 12 − 1 Spin-wave pole (Goldstone mode): R

LAD m0 2 χ (q, iω) , S↑↓(q, ω) = m δ(ω 2∆Bq ) ↑↓ ≈ −iω 2∆Bq2 0 − − However: Can perturbation theory be trusted?

10/22 ferromagnetic Luttinger liquid

in ferromagnetic phase: marginal couplings: dominant forward scattering effective low-energy theory in symmetry broken phase ⇒ Λ dq α† α 1 Hêff = αvσqψˆ (q)ψˆ (q) + fijρî( q)ρˆj (q) Z 2π σ σ 2L − Xασ −Λ Xq,ij

with renormalized couplings and densities Λ dq0 ˆα† 0 ˆα 0 ρˆn(q) = ασ −Λ 2π ψσ (q )ψσ (q + q). P R need Wilsonian RG to arrive at this model • irrelevant couplings stablize ferromagnetism! calculate correlation functions using bosonization • closed loop theorem (Dzyaloshinskii, Larkin, 1974) : • all corrections beyond RPA for χmm(q, ω) cancel! what happens to transverse spin waves in Luttinger liquid? •

11/22 Single particle Green function

2 ηn/2 2 ηm/2 1 r0 r0 Gσ(x, τ) = 2 2 2 2 2 2 2πi x + vnτ x + vmτ eiαkσx × [αx + iv τ]1/2[αx + iv τ]1/2 Xα n m

Luttinger liquid behavior • anomalous dimensions η = 1 (K + K−1 2) • i 4 i i − with K = [I + 1]−1/2 and K = [2(I 1)]−1/2 n m − η diverges for I 1 • m → Luttinger liquid relation: K = πv χ /2 • m m m at quantum phase transition: χ and v 0 m → ∞ m →

12/22 Transverse susceptibility from bosonization

¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¢£¤¥ ¦§ ¡ ¡ ¡ ¡ ¡ BOS BOS ¡ ¡ ¡ ¡ ¡ χ↑↓ (x, τ) ¡ ¡ ¡ ¡ ¡ ≡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡

m iα k k x r2 2η ( ↑− ↓) = −1 0 e (2π)2 x2+vm2 τ 2 α [αx+ivmτ]2 h i P no charge contribution • spin contribution is due to broken symmetry • anomalous scaling exponent 2η • m based on a linearized energy dispersion • only good for q k↑ k↓ • ≈ −

13/22 Low energy spin excitations

BOS S (q, ω) = C Θ(ω v q k↑ + k↓ ) ↑↓ m − m|| | − | 2ηm−1 [ω v ( q k↑ + k↓)] × − m | | − 2ηm+1 [ω + v ( q k↑ + k↓)] × m | | −

−1 4ηm with Cm = [4πvmΓ(2ηm)Γ(2 + 2ηm)] (r0/vm)

¡£¢ ¤ 1d Stoner ¢ •

continuum ¢

no Landau © ¦

•

damping © ¦

¥§¦ ¥§© ¤ ¤ ¨ ¢

14/22 Conclusions and outlook:

Eﬀective low-energy theory of weak ferromagnetic Luttinger liquids • Stoner parameter δ 2(I 1) measures distance to quantum critical point • ≈ − longitudinal susceptibility: SRPA(q, ω) = Z q δ(ω v q ) • m m| | − m| | v δ1/2, Z δ−1/2 χ Z /v δ−1 m ∝ m ∝ ⇒ m ∝ m m ∝ Luttinger liquid relation χ = 2K /πv • m m m anomalous dimension: η = 1 (K + K−1 2) at quantum ⇒ m 4 m m − → ∞ phase transition! Open problems: study ferromagnetic instability beyond Hartree-Fock theory • (numerics, Onsager correction) use Wilsonian renormalization group to study instability: • how do irrelevant operators (band curvature) stabilize ferromagnetism? formulation of problem as self-consistent determination of the Fermi • surface at the RG ﬁxed point.

15/22 Beyond Hartree-Fock: functional RG

Strategy: determine Fermi-surface (i.e. k↑, k↓) self-consistently as ﬁxed point of renormalization group! RG for fermions: 1d: g-ology [Solyom, 1979] • one-loop, using naive momentum-shell [Shankar (1994)] • functional RG, one-loop, using Polchinski equation: [Zanchi, Schulz • (1996); Salmhofer, Honerkamp (1998); Halboth, Metzner (1999)] using irreducible vertices [Honerkamp, Salmhofer (2001); P.K., Busche • (2001)] What is the problem with Fermions? Propagator singular on entire surface in k-space:

Zk G(k, ω) F ≈ ω v (k k ) iδ − F · − F

16/22 Fermi surface = RG ﬁxed point

Nozi`eres, 1964: • “In practice, we shall never try to calculate the Fermi surface, which is much too diﬃcult. What is more important to us is to know that it exists.”

Here: need renormalized Fermi surface k , k . • ↑ ↓ Strategy (S. Ledowski, P.K., J.Phys. Condens.Matter, 2003): • Requirement that RG ﬂow approaches ﬁxed point yields self-consistency condition for the renormalized Fermi surface! Implementation via functional RG for irreducible vertices •

exact ﬂow equation of two-point vertex Γ(2)(K; K) = [Σ (K) Σ(k , i0)] Λ − Λ − F

(2) δ( k0 kF Λ) (4) 0 0 ∂ΛΓΛ (K; K) = | − | − 0 ΓΛ (K, K ; K , K) Z 0 iω 0 k0 + µ Σ (K ) K n − − Λ

17/22 Flow of four- and six-point vertices

1 1 1 1 1 1 4 6 4 4 2 2 2 2 2 2 BCS

1 1 1 1 4 4

4 4 2 2 ZS 2 2

2 1 2 1 4 4

4 4 2 1 ZS 2 1

1 1 1 1 1 1 2 6 2 2 8 2 3 2 6 2 4 3 3 3 3 3 3

1 1 1 1 1 1 2 6 2 2 6 2 3 4 2 6 2 9 3 3 4 4 3 3 3 3

1 1 1 1 4 4 2 2 2 2 4 4 4 4 9 3 3 3 3

1 1 1 1 4 4 2 2 2 3 4 4 3 3 36 4 4

2 3 18/22 Scaling toward the Fermi surface

Field rescaling consider RG ﬂow of renormalized vertices: ⇐⇒

˜(2n) 0 0 Γl (Q1, . . . , Qn; Qn, . . . , Q1) = n0 n0 1/2 n−1 n−2 ˆ1 ˆn nˆn nˆ1 (2n) 0 0 ν0 Λ Zl Zl Zl Zl ΓΛ (K1, . . . , Kn; Kn, . . . , K1) . · · · · · · label momenta by unit vector nˆ and distance p from Fermi surface: k = kF + vˆF p v p scaling variables: Q = (nˆ, pξ, iωτ) k F kF length: ξ = vF /Λ ; time: τ = 1/Λ n infrared cutoﬀ Λ (units of energy)

The renormalized vertices can be viewed as dimensionless dynamic scaling functions! ˜(2) example: 2-point vertex Γl (Q; Q) = Φl(nˆ, pξ, ωτ)

19/22 ﬂow equation for FS shift

Znˆ rescaled two-point vertex Γ˜(2)(Q) = l [Σ (K) Σ(k , i0)] satisﬁes l − Λ Λ − F

˜(2) nˆ ˜(2) ˙ 0 ˜(4) 0 0 ∂lΓl (Q) = (1 ηl q∂q ∂)Γl (Q) Gl(Q )Γl (Q, Q ; Q , Q) − − − − ZQ0

relevant couplings µ˜nˆ Γ˜(2)(σ, nˆ, q = 0, i = i0) = Γ˜(2)(Q ) satisfy l ≡ l l 0 n n n ∂ µ˜ˆ = (1 ηˆ )µ˜ˆ + Γ˙ (2)(nˆ) l l − l l l

(2) 0 (4) 0 0 with Γ˙ (nˆ) = 0 G˙ (Q )Γ˜ (Q , Q ; Q , Q ). Equivalent integral equation: l − Q l l 0 0 R

l n l nˆ n t nˆ (2) ˆ l− 0 dτητ ˆ −t+ 0 dτητ ˙ µ˜l = e µ˜0 + dte Γt (nˆ) Z0

nˆ nˆ Want: RG ﬁxed point: µ˜∞ = liml→∞ µ˜l ﬁnite!

20/22 exact integral equation for FS

∞ n l nˆ (4) ˆ −l+ 0 dτητ ˙ 0 ˜ 0 0 µ˜0 = dle Gt(Q )Γl (Q0, Q ; Q , Q0) Z0 ZQ0

Implicit equation for µñˆ , relating it to the values of the two-point vertex • 0 and the four-point vertex on the entire RG trajectory! Initial value fixes counterterm: Σ(k , i0) Σ (k , i0) = ξ µñˆ /Znˆ . • F − ξ0 F 0 0 0 Alternative form (Σ (k , i0) = 0 and Z = 1, ξk0 = k0 k0 .) • Λ0 F 0 | − F | 0 0 dk dω Θ(Λ ξk0 ) k 0 Σ( F , i0) = D 0 − 0 Z (2π) 2π iω k0 + µ Σξ 0 (K ) Xσ0 − − k (4) 0 0 0 0 0 0 Γ (kF , i0, σ, k , iω , σ ; k , iω , σ , kF , i0, σ) × ξk0

in simplest approximation: self-consistent Hartree-Fock: • dk0 (4) 0 0 0 Σ(k , i0) = 0 D Γ (k , σ; k , σ )Θ (µ k0 Σ(k , i0)) F σ (2π) 0 F F − − F P R 21/22 Open problems, work in progress:

use exact integral equation for Fermi surface to calculate corrections to • Hartree-Fock RG-ﬂow: how do irrelevant couplings stabilize ferromagnetic ground state • in 1d single-particle properties near ferromagnetic quantum critical point in 2d? •

22/22